Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

Combinatorial Invariants of Toric Arrangements

An arrangement is a collection of subspaces of a topological space. For example, a set of codimension one affine subspaces in a finite dimensional vector space is an arrangement of hyperplanes. A general question in arrangement theory is to determine to what extent the combinatorial data of an arrangement determines the topology of the complement of the arrangement. Established combinatorial structures in this context are matroids and -for hyperplane arrangements in the real vector space- oriented matroids. Let X be the punctured plane C- 0 or the unit circle S 1, and a(1),...,a(n) integer vectors in Z d. By interpreting the a(i) as characters of the torus T=Hom(Z d,X) isomorphic to X d we obtain a toric arrangement in T by considering the set of kernels of the characters. A toric arrangement is covered naturally by a periodic affine hyperplane arrangement in the d-dimensional complex or real vector space V=C d or R d (according to whether X = C- 0 or S 1). Moreover, if V is the real vector space R d the stratification of V given by a finite hyperplane arrangement can be combinatorially characterized by an affine oriented matroid. Our main objective is to find an abstract combinatorial description for the stratification of T given by the toric arrangement in the case X=S 1 - and to develop a concept of toric oriented matroids as an abstract characterization of arrangements of topological subtori in the compact torus (S 1) d. Part of our motivation comes from the possible generalization of known topological results about the complement of complexified toric arrangements to such toric pseudoarrangements. Towards this goal, we study abstract combinatorial descriptions of locally finite hyperplane arrangements and group actions thereon. First, we generalize the theory of semimatroids and geometric semilattices to the case of an infinite ground set, and study their quotients under group actions from an enumerative and structural point of view. As a second step, we consider corresponding generalizations of affine oriented matroids in order to characterize the stratification of R d given by a locally finite non-central arrangement in R d in terms of sign vectors

Vanishing results for the cohomology of complex toric hyperplane complements

Suppose \Cal R is the complement of an essential arrangement of toric hyperlanes in the complex torus (\C^*)^n and \pi=\pi_1(\Cal R). We show that H^*(\Cal R;A) vanishes except in the top degree $n$ when $A$ is one of the following systems of local coefficients: (a) a system of nonresonant coefficients in a complex line bundle, (b) the von Neumann algebra \cn\pi, or (c) the group ring \zz \pi. In case (a) the dimension of $H^n$ is |e(\Cal R)| where e(\Cal R) denotes the Euler characteristic, and in case (b) the $n^{\mathrm{th}}$ \eltwo Betti number is also |e(\Cal R)|.Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:math/061240

The homotopy type of toric arrangements

A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group we also provide an algebraic description, very handy for cohomology computations. In the last part we give a description in terms of tableaux for a toric arrangement appearing in robotics.Comment: To appear on J. of Pure and Appl. Algebra. 16 pages, 3 picture

The maximum likelihood degree of a very affine variety

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern-Schwartz-MacPherson class. The strengthened version recovers the geometric deletion-restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko's theorem on the Newton polytope for nondegenerate hypersurfaces.Comment: Improved readability. Final version, to appear in Compositio Mathematic